The relationships (also termed “trends”) between petrophysical properties such as porosity, permeability, formation factor, elastic properties, relative permeability, and capillary pressure, are useful for various geological and engineering applications (Nelson, 1994), and are regarded as crucial to the to accurate characterization and evaluation of rocks/reservoirs, by which a thorough understanding is achieved. However, many factors such as degree of heterogeneity, rock formation, pore geometry, grain size, packing and solution/dissolution, cause the trends to vary in a complex manner (Ma and Morrow, 1996).
Scientists and engineers have employed various experimental approaches to establish trends (see, e.g. Ma & Morrow, 1996, Ehrenberg & Nadeau, 2005, Weibel et al., 2012, Vik et al., 2013, Torabi et al., 2013). The data resulting from these approaches generally has a large amount of scatter and deviation that make it difficult to discern any well-defined trends (Weibel et al., 2012). Additionally, such experiments often require weeks and large number of samples to establish a statistically meaningful trend, and consequently are vulnerable to experimental errors and difficulties.
Such issues can be avoided with the use of digital rock physics (DRP), which employs advanced imaging technologies, such as microscopy and spectroscopy, to construct a digital representation of the rock or other material at a chosen level of magnification and resolution. The digital representation includes, but is not limited to, a two- or three-dimensional image of a sample of the material. Computerized analysis techniques may then be applied to the acquired image to visualize the internal structure and/or to characterize the material. Depending on the analysis, a number of characteristic properties are measured, quantified, and inter-related. Even in the absence of experimental error, however, existing analysis techniques fail to suitably account for heterogeneities and other complicating factors that make it difficult to discern meaningful trends.
Moreover, while some useful characterization and conclusions may be derived from analysis of samples that can be directly magnified and imaged, the scale of a reservoir and its component formations is much too large to be directly imaged and analyzed. Nor is it feasible to perform a sufficient number of experiments on a large enough scale to extract trend information. Yet the importance of such large scale trend information to accurate reservoir evaluation and forecasting cannot be overemphasized. The main complicating factor for the determination of such trend information is the high degree of structural heterogeneity that is present in most reservoir rocks (Worthington, 2004), i.e. such rocks include more than one typical pore size and structure.
Numerous upscaling techniques for predicting large scale petrophysical properties from sample-derived trend information have been presented in the literature. A majority of these techniques are restricted to the study of the single-phase permeability of a porous material. For example, Durlofsky (Durlofsky, 2005) compared a variety of approaches for gridding and upscaling geocellular models for flow simulation. Khalili et al. (Khalili et al., 2012) established porosity transforms between high-resolution (small scale) and low-resolution (large scale) images to calibrate a low resolution porosity map, which can then be used to populate permeability on the low-resolution image. Renormalization schemes for upscaling have been proposed by Green & Paterson (Green & Paterson, 2007) and Krabbenhoft & Karim (Krabbenhoft & Karim, 2010). The conclusion of each of the studies above are similar: the results are sensitive to property contrasts, i.e., the range between the lowest and highest values, which depends strongly on the degree of heterogeneity of the porous material. More importantly, they found that the translation between large-scale and small-scale sample permeability varies greatly from sample to sample. In at least some cases, the large-scale sample tends to have a higher permeability that would be predicted by the small scale sample (Ehrenberg, 2007). Clearly, the existing methods fail to sufficiently resolve the relevant petrophysical properties in a manner suitable for upscaling.